I believe the webapp retrieves your message history from the phone in real time, and Whatsapp does not store messages on their servers once they have been delivered.
No. If you take the example animation, you can set border-left-color and border-right-color to transparent, and set border-top-width to 0. This will place the triangle in a bounding box of the correct width and height.
Another great example http://www.nytimes.com/interactive/2013/05/25/sunday-review/...